Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{-2t^2 + 16t + 18}{8t^2 - 8t - 16}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ p = \dfrac {-2(t^2 - 8t - 9)} {8(t^2 - t - 2)} $ $ p = -\dfrac{2}{8} \cdot \dfrac{t^2 - 8t - 9}{t^2 - t - 2} $ Simplify: $ p = - \dfrac{1}{4} \cdot \dfrac{t^2 - 8t - 9}{t^2 - t - 2}$ Next factor the numerator and denominator. $ p = - \dfrac{1}{4} \cdot \dfrac{(t + 1)(t - 9)}{(t + 1)(t - 2)}$ Assuming $t \neq -1$ , we can cancel the $t + 1$ $ p = - \dfrac{1}{4} \cdot \dfrac{t - 9}{t - 2}$ Therefore: $ p = \dfrac{ -t + 9 }{ 4(t - 2)}$, $t \neq -1$